Recently, there has been the so-called Hubble tension in cosmology, indicating a mismatch between the local measurements and early-time observations for the expansion rate of the Universe, i.e. the Hubble parameter \(H_0\) . The measurements from the Hubble Space Telescope (HST) has implied the value of \(H_0\)  to be \(74.03 \pm 1.42\)  \(\rm km\;s^{-1}Mpc^{-1}\)  [1]}, whereas, the CMB measurement together with the \(\Lambda CDM\)  model has given \(67.4 \pm 0.5\)  \(\rm km\;s^{-1}Mpc^{-1}\)  [2]}.
Many approaches in the literature have been presented to understand this tension, which can be divided into the early-time and late-time modifications
of general relativity. Examples for the former are the early dark energy scenario [3]}, [4]}, primodial magnetic fields [5]}, dark energy-dark matter interactions [6]}, while the latter includes the modified gravity theory, which is our adopted approach.
In this study, we concentrate on one of the simplest dark energy model from the generalized Proca theory, in which we only consider up to the cubic order of the vector field self-interactions
and present the numerical analysis of the model based on
Refs. [7]}.
